Solving For X: Linear Pairs And Angle Measures

Hey math enthusiasts! Today, we're diving into a classic geometry problem that's all about angles, linear pairs, and a bit of algebra. The question is: If the measures of two adjacent angles are $(4x+10)^{\circ}$ and $(x+20)^{\circ}$ and they form a linear pair, then the value of $x$ is? We'll break it down step by step, making sure everyone understands how to crack this type of problem. So, grab your pencils, and let's get started!

Understanding the Basics: Linear Pairs

First things first, what exactly is a linear pair? Well, a linear pair is simply two adjacent angles that form a straight line. And what do we know about straight lines in terms of angles? They always add up to $180^{\circ}$! This is the crucial piece of information we'll use to solve our problem. Think of it like this: a straight line is a flat angle, and a flat angle measures $180^{\circ}$. When two angles come together to form that flat angle, their measures must combine to equal $180^{\circ}$. So, the core concept here is: A linear pair of angles always adds up to 180 degrees.

Imagine two slices of pizza next to each other, perfectly filling the whole pizza. If the pizza is a straight line (180 degrees), then the two slices (angles) must add up to fill the whole pizza. This fundamental concept is the key to unlocking our problem. Now, let's look at the specific angles given in our question. We have two angles: $(4x + 10)^{\circ}$ and $(x + 20)^{\circ}$. The problem states that these two angles form a linear pair. Therefore, we know that the sum of their measures is equal to $180^{\circ}$. This gives us the foundation for setting up our equation. This understanding is fundamental to mastering geometry problems involving angles and will be used repeatedly, so take a moment to be certain you are following the steps.

Setting Up the Equation

Okay, guys, now that we know the drill, let's get down to business and set up our equation. Since the two angles form a linear pair, their measures add up to $180^{\circ}$. So, we can write the equation like this:

$(4x + 10) + (x + 20) = 180$

See? It's all about translating the words into mathematical symbols. The problem says they form a linear pair, which means they add up to 180. We just put the expressions representing the angle measures on the left side of the equation and the value of a straight angle (180) on the right side.

Now, let's simplify this equation. First, combine like terms on the left side. We have $4x$ and $x$, which combine to make $5x$. And we have $10$ and $20$, which add up to $30$. So, our equation becomes:

$5x + 30 = 180$

This is a much cleaner, simpler equation that's easier to solve. We've successfully turned a word problem into a manageable algebraic equation. You'll notice, this is much more basic than it originally looks, and we have gotten one step closer to finding our solution. This step is very important because, with complicated equations, one small error can be multiplied, so it's important to make sure your equation is correct and that you are ready to move forward.

Solving for x

Alright, time to solve for x! We have the simplified equation: $5x + 30 = 180$. To isolate x, we need to get rid of the $30$ first. We can do this by subtracting $30$ from both sides of the equation. Remember, whatever you do to one side of the equation, you must do to the other side to keep things balanced. This is the golden rule of algebra! So, subtracting $30$ from both sides, we get:

$5x + 30 - 30 = 180 - 30$

This simplifies to:

$5x = 150$

Now, we need to get x completely by itself. Currently, it's being multiplied by $5$. To undo that, we divide both sides of the equation by $5$:

rac{5x}{5} = rac{150}{5}

This gives us:

$x = 30$

And there you have it! We've solved for x. The value of $x$ is $30$. Not so bad, right? We have successfully navigated the algebraic steps to arrive at our answer. Now, let's take a moment to check our work and make sure this value makes sense in the context of the original problem.

Checking Our Solution

It's always a good idea to check your work. Let's plug the value of $x = 30$ back into the original expressions for the angles to see if they indeed form a linear pair. The first angle is $(4x + 10)^{\circ}$. Substituting $x = 30$, we get:

$(4 imes 30 + 10)^{\circ} = (120 + 10)^{\circ} = 130^{\circ}$

The second angle is $(x + 20)^{\circ}$. Substituting $x = 30$, we get:

$(30 + 20)^{\circ} = 50^{\circ}$

Now, let's add these two angles together: $130^{\circ} + 50^{\circ} = 180^{\circ}$. And what do you know? They add up to $180^{\circ}$, which confirms that they form a linear pair! This means our solution, $x = 30$, is correct. We have successfully solved the problem and validated our answer. This is a crucial step because it not only confirms our solution but also reinforces our understanding of the concepts. Always take the time to double-check your work; it is an important part of problem-solving.

Choosing the Correct Answer

Going back to the multiple-choice options, we see that our answer, $x = 30$, corresponds to option A. So, the correct answer is A. $30^{\circ}$. Congrats, you've successfully navigated the problem!

Recap and Tips

Here’s a quick recap of the key steps:

1. Understand the concept: Remember that a linear pair of angles adds up to $180^{\circ}$.
2. Set up the equation: Translate the problem into a mathematical equation.
3. Simplify and solve: Combine like terms and solve for x.
4. Check your work: Plug the value of x back into the original expressions to make sure they form a linear pair.

• Practice, practice, practice: The more you solve these types of problems, the easier they will become. Try different examples and variations to build your confidence.
• Draw diagrams: Visualizing the problem with a diagram can help you understand the relationships between the angles.
• Break it down: Don't get overwhelmed! Break the problem down into smaller, manageable steps.

I hope this helped you better understand how to solve this type of problem. Keep practicing, and you'll become a geometry pro in no time. Keep learning, keep practicing, and you’ll be amazed at how quickly you improve.

Additional Practice Problems

To solidify your understanding, try these practice problems:

1. If two adjacent angles form a linear pair and one angle measures $65^{\circ}$, what is the measure of the other angle?
2. Two angles form a linear pair. One angle is represented by $2x + 10$ and the other by $x - 40$. Find the value of $x$ and the measure of each angle.
3. The measures of two angles forming a linear pair are in the ratio of 2:3. Find the measure of each angle.

Good luck, and keep up the great work!